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In this paper, we study the supercharacter theories of elementary abelian $p$-groups of order $p^{2}$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^{2}$ that has a non-identity superclass of size $1$ or a non-principal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2,\, 3,\, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.
Let
$\eta (G)$
be the number of conjugacy classes of maximal cyclic subgroups of G. We prove that if G is a p-group of order
$p^n$
and nilpotence class l, then
$\eta (G)$
is bounded below by a linear function in
$n/l$
.
Morgan and Parker proved that if G is a group with
${\textbf{Z}(G)} = 1$
, then the connected components of the commuting graph of G have diameter at most
$10$
. Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a
$2$
-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most
$8$
. We prove that the hypothesis
$Z (G) = 1$
in these results can be replaced with
$G' \cap {\textbf{Z}(G)} = 1$
. We also prove that if G is solvable and
$G/{\textbf{Z}(G)}$
is either a Frobenius group or a
$2$
-Frobenius group, then the commuting graph of G is disconnected.
The first demonstration of laser action in ruby was made in 1960 by T. H. Maiman of Hughes Research Laboratories, USA. Many laboratories worldwide began the search for lasers using different materials, operating at different wavelengths. In the UK, academia, industry and the central laboratories took up the challenge from the earliest days to develop these systems for a broad range of applications. This historical review looks at the contribution the UK has made to the advancement of the technology, the development of systems and components and their exploitation over the last 60 years.
For a group G, we define a graph
$\Delta (G)$
by letting
$G^{\scriptsize\#}=G\setminus {\{\,1\,\}} $
be the set of vertices and by drawing an edge between distinct elements
$x,y\in G^{\scriptsize\#}$
if and only if the subgroup
$\langle x,y\rangle $
is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate
$\Delta (G)$
for a Z-group G.
Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.
Stars form in clusters, while planets form in gaseous disks around young stars. Cluster dissolution occurs on longer time scales than disk dispersal. Planet formation thus typically takes place while the host star is still inside the cluster. We explore how the presence of other stars affects the evolution of circumstellar disks. Our numerical approach requires multi-scale and multi-physics simulations where the relevant components and their interactions are resolved. The simulations start with the collapse of a turbulent cloud, from which stars with disks form, which are able to influence each other. We focus on the effect of extinction due to residual cloud gas on the early evolution of circumstellar disks. We find that this extinction protects circumstellar disks against external photoevaporation, but these disks then become vulnerable to dynamic truncation by passing stars. We conclude that circumstellar disk evolution is heavily affected by the early evolution of the cluster.
We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.
Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$-subgroup of $G$. We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.
Let $G$ be a $p$-group and let $\unicode[STIX]{x1D712}$ be an irreducible character of $G$. The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^{n}$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$, then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.
Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.
Let $G$ be a finite group and let $p$ be a prime factor of $|G|$. Suppose that $G$ is solvable and $P$ is a Sylow $p$-subgroup of $G$. In this note, we prove that $P{\vartriangleleft}G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all irreducible monomial $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.
Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).
Let $G$ be a finite solvable group and let $p$ be a prime. We prove that the intersection of the kernels of irreducible monomial $p$-Brauer characters of $G$ with degrees divisible by $p$ is $p$-closed.
Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$-Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.
When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.
We define several graphs related to the p-blocks of a solvable group. We bound the diameter of these graphs when the defect group associated with the block is either abelian or normal and when the group has odd order. We give examples to show that these bounds are met.